Skip to main content
SpringerLink
Log in

Extended Projection Methods for Monotone Variational Inequalities

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Harker, P. T., and Pang, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 1–60, 1990.

    Google Scholar 

  2. Pang, J. S., and Chan, D., Iterative Methods for Variational and Complementarity Problems, Mathematical Programming, Vol. 24, pp. 248–313, 1982.

    Google Scholar 

  3. Dafermos, S., Traffic Equilibrium and Variational Inequalities, Transportation Science, Vol. 14, pp. 42–54, 1980.

    Google Scholar 

  4. Dafermos, S., An Iterative Scheme for Variational Inequalities, Mathematical Programming, Vol. 26, pp. 40–47, 1983.

    Google Scholar 

  5. Fukushima, M., Equivalent Differentiable Optimization Problems and Descent Method for Asymmetric Variational Inequality Problems, Mathematical Programming, Vol. 53, pp. 99–110, 1992.

    Google Scholar 

  6. Marcotte, P., and Wu, J. H., On the Convergence of Projection Methods: Application to the Decomposition of Affine Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 85, pp. 363–376, 1995.

    Google Scholar 

  7. Bertsekas, D. P., and Tsitsiklis, J., Parallel and Distributed Computation: Numerical Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.

    Google Scholar 

  8. Korpelevich, G. M., The Extragradient Method for Finding Saddle Points and Other Problems, Matecon, Vol. 12, pp. 747–756, 1976.

    Google Scholar 

  9. Khobotov, E. N., Modification of the Extragradient Method for the Solution of Variational Inequalities and Some Optimization Problems, USSR Computational Mathematics and Physics, Vol. 27, pp. 120–127, 1987.

    Google Scholar 

  10. Marcotte, P., Application of Khobotov's Algorithm to Variational Inequalities and Network Equilibrium Problems, Information Systems and Operations Research, Vol. 29, pp. 258–270, 1991.

    Google Scholar 

  11. Solodov, M. V., and Tseng, P., Modified Projection-Type Methods for Monotone Variational Inequalities, SIAM Journal on Control and Optimization, Vol. 34, pp. 1814–1830, 1996.

    Google Scholar 

  12. Sun, D., A Class of Iterative Methods for Solving Nonlinear Projection Equations, Journal of Optimization Theory and Applications, Vol. 91, pp. 123–140, 1996.

    Google Scholar 

  13. He, B., and Stoer, J., Solutions of Projection Problems over Polytopes, Numerische Mathematik, Vol. 61, pp. 73–90, 1992.

    Google Scholar 

  14. He, B., A New Method for a Class of Linear Variational Inequalities, Mathematical Programming, Vol. 66, pp. 137–144, 1994.

    Google Scholar 

  15. He, B., A Projection and Contraction Method for a Class of Linear Complementarity Problems and Its Application in Convex Quadratic Programming, Applied Mathematics and Optimization, Vol. 25, pp. 247–262, 1992.

    Google Scholar 

  16. He, B., On a Class of Iterative Projection and Contraction Methods for Linear Programming, Journal of Optimization Theory and Applications, Vol. 78, pp. 247–266, 1993.

    Google Scholar 

  17. Noor, M. A., An Iterative Scheme for a Class of Quasi-Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 110, pp 463–468, 1985.

    Google Scholar 

  18. Noor, M. A., Quasi-Variational Inequalities, Applied Mathematics Letters, Vol. 1, pp. 367–370, 1988.

    Google Scholar 

  19. Douglas, J., and Rachford, H. H., On the Numerical Solution of the Heat Conduction Problem in 2 and 3 Space Variables, Transactions of the American Mathematical Society, Vol. 82, pp. 421–439, 1956.

    Google Scholar 

  20. Lions, P. L., and Mercier, B., Splitting Algorithms for the Sum of Two Nonlinear Operators, SIAM Journal on Numerical Analysis, Vol. 16, pp. 964–979, 1979.

    Google Scholar 

  21. Ortega, J. M., and Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, New York, 1970.

    Google Scholar 

  22. Ratschek, H., and Rokne, J., Computer Methods for the Range of Functions, Ellis Horwood, Chichester, England, 1984.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, P. R. China

    Y. B. Zhao (Researcher)

Authors
  1. Y. B. Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, Y.B. Extended Projection Methods for Monotone Variational Inequalities. Journal of Optimization Theory and Applications 100, 219–231 (1999). https://doi.org/10.1023/A:1021781318170

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021781318170

Search

Navigation